Local search methods based on variable focusing for random K-satisfiability

We introduce variable focused local search algorithms for satisfiabiliity problems. Usual approaches focus uniformly on unsatisfied clauses. The methods described here work by focusing on random variables in unsatisfied clauses. Variants are considered where variables are selected uniformly and randomly or by introducing a bias towards picking variables participating in several unsatistified clauses. These are studied in the case of the random 3-SAT problem, together with an alternative energy definition, the number of variables in unsatisfied constraints. The variable-based focused Metropolis search (V-FMS) is found to be quite close in performance to the standard clause-based FMS at optimal noise. At infinite noise, instead, the threshold for the linearity of solution times with instance size is improved by picking preferably variables in several UNSAT clauses. Consequences for algorithmic design are discussed.